Flat Loss, Evolving Features: Grokking in Modular Addition

Grokking the delayed transition from memorization to generalization in neural networks remains poorly understood, in part because prior empirical studies confound the roles of architecture, optimization, and regularization. We present a controlled study that systematically disentangles these factors on modular addition (mod 97), with matched and carefully tuned training regimes across models. Our central finding is that grokking dynamics are not primarily determined by architecture, but by interactions between optimization stability and regularization. Specifically, we show: (1) depth has a non-monotonic effect, with depth-4 MLPs consistently failing to grok while depth-8 residual networks recover generalization, demonstrating that depth requires architectural stabilization; (2) the apparent gap between Transformers and MLPs largely disappears (1.11$\times$ delay) under matched hyperparameters, indicating that previously reported differences are largely due to optimizer and regularization confounds; (3) activation function effects are regime-dependent, with GELU up to 4.3$\times$ faster than ReLU only when regularization permits memorization; and (4) weight decay is the dominant control parameter, exhibiting a narrow ``Goldilocks'' regime in which grokking occurs, while too little or too much prevents generalization. Across 3--5 seeds per configuration, these results provide a unified empirical account of grokking as an interaction-driven phenomenon. Our findings challenge architecture-centric interpretations and clarify how optimization and regularization jointly govern delayed generalization.

Grokking-the phenomenon where validation accuracy of neural networks on modular addition of two integers rises long after training data has been memorized-has been characterized in previous works as producing sinusoidal input weight distributions in transformers and multi-layer perceptrons (MLPs). We find empirically that ReLU MLPs in our experimental setting instead learn near-binary square wave input weights, where intermediate-valued weights appear exclusively near sign-change boundaries, alongside output weight distributions whose dominant Fourier phases satisfy a phase-sum relation $φ_{\mathrm{out}} = φ_a + φ_b$; this relation holds even when the model is trained on noisy data and fails to grok. We extract the frequency and phase of each neuron's weights via DFT and construct an idealized MLP: Input weights are replaced by perfect binary square waves and output weights by cosines, both parametrized by the frequencies, phases, and amplitudes extracted from the dominant Fourier components of the real model weights. This idealized model achieves 95.5% accuracy when the frequencies and phases are extracted from the weights of a model trained on noisy data that itself achieves only 0.23% accuracy. This suggests that grokking does not discover the correct algorithm, but rather sharpens an algorithm substantially encoded during memorization, progressively binarizing the input weights into cleaner square waves and aligning the output weights, until generalization becomes possible.

In grokking an early fit to the training data separates from a much later improvement in generalization. During this delay, training can move from a fixed neural tangent kernel (NTK) regime to one in which task-relevant kernel eigendirections continue to evolve. We provide a quantitative theory for how this transition from lazy to rich learning can produce delayed generalization. For homogeneous networks trained with squared loss and $L_2$ weight decay, we show that a finite residual remains after memorization, with larger residual fractions in target components associated with smaller NTK eigenvalues. These residuals feed back into the dynamics of the NTK itself, and projecting the resulting dynamics onto task-relevant spectral directions yields a reduced system in which residual-driven kernel growth competes with weight decay. This system predicts that the grokking timescale is controlled by the product of learning rate and weight decay, that feature learning slows logarithmically near a critical decay above which task-aligned NTK structure can no longer support generalization, and that stronger decay can prevent fitting altogether. We test these predictions in modular addition. In a homogeneous MLP, task-aligned Fourier structure continues to emerge in the NTK after training accuracy has saturated, and an 84$\times$90-grid of trained networks across varying learning rate and weight decay recovers the predicted phase geometry and inverse-product scaling of the generalization time with learning rate and weight decay. A one-block Transformer shows similar macroscopic phase structure in a 42$\times$45-grid, as well as the same transition-time scaling despite violating exact homogeneity. Together, these results provide a mechanistic derivation connecting post-fit feature learning to both the onset of generalization and its phase structure in the learning rate and weight decay plane.